Artin–Tits group, pure subgroup and positive lifts

PDF p. 5: §2.2 after (2.4)

Let $(W,S)$ be a Coxeter system with $S=\{s_i:i\in {\mathrm{\Gamma }}_0\}$ finite, and let ${\mathcal{X}}^{\mathbb{C}}$ be its classical complexified hyperplane complement. Choose $Z\in {\mathcal{X}}^{\mathbb{C}}$ and write $\bar{Z}$ for its orbit in ${\mathcal{X}}^{\mathbb{C}}\mathrm{/}W$. The Artin–Tits group is $A={\pi }_1({\mathcal{X}}^{\mathbb{C}}\mathrm{/}W,\bar{Z})$, the group of based loop-homotopy classes in the quotient. The pure Artin–Tits group is $P={\pi }_1({\mathcal{X}}^{\mathbb{C}},Z)$, identified with its image in $A$ under the covering map. Let $\pi :A\to W$ send a loop class to the unique $w$ for which its lift starting at $Z$ ends at $wZ$.

Fix an isomorphism over $W$ from the presented Artin group to $A$ supplied by the fundamental-group identification. The presented group has generators $a_s$ for $s\in S$ and, for distinct $s,t$ with finite $m(s,t)=\mathrm{ord}(st)$, the relation equating the two alternating words of length $m(s,t)$ beginning with $a_s$ and $a_t$ respectively. Write ${\sigma }_s\in A$ for the image of $a_s$, and ${\sigma }_i={\sigma }_{s_i}$. For a reduced expression $w=s_1\cdots s_r$, meaning an expression of minimum length among words in $S$ representing $w$, define the positive lift ${\sigma }_w={\sigma }_{s_1}\cdots {\sigma }_{s_r}$; the empty product is the identity. Thus $w\mapsto {\sigma }_w$ is a set-theoretic section of $\pi$.

The Coxeter-system supplier defines $W$, its generators and word length; the complement supplier defines ${\mathcal{X}}^{\mathbb{C}}$. Based fundamental groups consist of loop-homotopy classes with concatenation. The classical-cover result gives the covering, the endpoint map $\pi$, and the exact sequence $1\to P\to A\stackrel{\pi }{\to }W\to 1$ after identifying $P$ with its image. In particular, $P=\ker \pi$.

The independent fundamental-group identification applies to the finite-rank dual reflection system identified in the classical-cover result. It gives an isomorphism of the presented Artin-group sequence with this covering sequence over $W$; fixing one makes the displayed generators elements of $A$. The Artin presentation defines the alternating relations. The positive-lift interface proves that products corresponding to any two reduced expressions for $w$ agree and that their image under the projection is $w$. Transporting these facts through the fixed isomorphism makes ${\sigma }_w$ well defined and gives $\pi ({\sigma }_w)=w$. A set-theoretic section is a function whose composite with $\pi$ is the identity; no homomorphism property is asserted.